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Simplifying (28x + -7)(24x + 18) = 180 Reorder the terms: (-7 + 28x)(24x + 18) = 180 Reorder the terms: (-7 + 28x)(18 + 24x) = 180 Multiply (-7 + 28x) * (18 + 24x) (-7(18 + 24x) + 28x * (18 + 24x)) = 180 ((18 * -7 + 24x * -7) + 28x * (18 + 24x)) = 180 ((-126 + -168x) + 28x * (18 + 24x)) = 180 (-126 + -168x + (18 * 28x + 24x * 28x)) = 180 (-126 + -168x + (504x + 672x2)) = 180 Combine like terms: -168x + 504x = 336x (-126 + 336x + 672x2) = 180 Solving -126 + 336x + 672x2 = 180 Solving for variable 'x'. Reorder the terms: -126 + -180 + 336x + 672x2 = 180 + -180 Combine like terms: -126 + -180 = -306 -306 + 336x + 672x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -306 + 336x + 672x2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(-51 + 56x + 112x2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-51 + 56x + 112x2)' equal to zero and attempt to solve: Simplifying -51 + 56x + 112x2 = 0 Solving -51 + 56x + 112x2 = 0 Begin completing the square. Divide all terms by 112 the coefficient of the squared term: Divide each side by '112'. -0.4553571429 + 0.5x + x2 = 0 Move the constant term to the right: Add '0.4553571429' to each side of the equation. -0.4553571429 + 0.5x + 0.4553571429 + x2 = 0 + 0.4553571429 Reorder the terms: -0.4553571429 + 0.4553571429 + 0.5x + x2 = 0 + 0.4553571429 Combine like terms: -0.4553571429 + 0.4553571429 = 0.0000000000 0.0000000000 + 0.5x + x2 = 0 + 0.4553571429 0.5x + x2 = 0 + 0.4553571429 Combine like terms: 0 + 0.4553571429 = 0.4553571429 0.5x + x2 = 0.4553571429 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = 0.4553571429 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 0.4553571429 + 0.0625 Combine like terms: 0.4553571429 + 0.0625 = 0.5178571429 0.0625 + 0.5x + x2 = 0.5178571429 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 0.5178571429 Calculate the square root of the right side: 0.719622917 Break this problem into two subproblems by setting (x + 0.25) equal to 0.719622917 and -0.719622917.Subproblem 1
x + 0.25 = 0.719622917 Simplifying x + 0.25 = 0.719622917 Reorder the terms: 0.25 + x = 0.719622917 Solving 0.25 + x = 0.719622917 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 0.719622917 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 0.719622917 + -0.25 x = 0.719622917 + -0.25 Combine like terms: 0.719622917 + -0.25 = 0.469622917 x = 0.469622917 Simplifying x = 0.469622917Subproblem 2
x + 0.25 = -0.719622917 Simplifying x + 0.25 = -0.719622917 Reorder the terms: 0.25 + x = -0.719622917 Solving 0.25 + x = -0.719622917 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -0.719622917 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -0.719622917 + -0.25 x = -0.719622917 + -0.25 Combine like terms: -0.719622917 + -0.25 = -0.969622917 x = -0.969622917 Simplifying x = -0.969622917Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.469622917, -0.969622917}Solution
x = {0.469622917, -0.969622917}
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